Cancer maps can provide important clues concerning geographical variability in the etiology, prevention, screelling or treatment of cancer. As in all medical research, it ts important to determine whether any variation observed may reasonably be due to chance or not. This can be done using tests for spatial randomness, adjusting for the uneven geographical population density. Many such tests have been proposed, but for most, little is known about their properties, and they are seldomly used in cancer atlases. In this methodological project we will (i) develop theoretical properties that any test for spatial randomness should fulfill in order to be useful for cancer maps, (ii) determine which test statistics do and do not fulfifi these properties, (iii) evaluate the statistical power of different test statistics for different alternative hypotheses, (iv) determine the ability of different tests to estimate cluster model parameters when the null hypothesis is rejected, and (v) evaluate and ifiustrate the practical use of different test statistics on, among other data sets, county based brain cancer mortality data from the United States and individually geocoded breast cancer treatment data from Connecticut.